Riemannian manifolds are KKM spaces

Year 2019, Volume: 3 Issue: 2, 64 - 73, 30.06.2019

Sehie Park

https://doi.org/10.31197/atnaa.513857

Cited By: 2

Abstract

Let (M,g) be a complete, finite-dimensional Riemannian manifold. Based on the fact that any geodesic convex subset of M is a KKM space, we establish the KKM theory on such subsets originated from the Knaster-Kuratowski-Mazurkiewitz theorem in 1929.

Keywords

Riemannian manifold, KKM theorem, Abstract convex space, Geodesic convex set

References

• 1] M.A. Alghamdi, W.A. Kirk, and N. Shahzad, Locally nonexpansive mappings in geodesic and length spaces, Top. Appl. 173 (2014) 59–73. [2] Ariza-Ruiz, D., Li, C., and Lopez-Acedo, G. The Schauder fixed point theorem in geodesic spaces, J. Math. Anal. Appl. 417 (2014) 345–360. [3] Chaipunya, P. and Kumam, P. Nonself KKM maps and corresponding theorems in Hadamard manifolds, Appl. Gen. Topol. 16(1) (2015) 37-44. [4] Chen, S.-L., Huang, N.-J., and OoRegan, D. Geodesic B-preinvex functions and multiobjective optimization problems on Riemannian manifolds, J. Appl. Math. Vol.2014, Article ID 524698, 12 pages. doi.org/10.1155/2014/524698 [5] Colao, V., Lopez, G., Marino, G., and Martin-Marquez,V. Equilibrium problems in Hadamard manifolds, J. Math. Anal. Appl. 388 (2012) 61–77. [6] Cruz Neto, J.X., Jacinto, F.M.O., Soares, P.A. Jr., and Souza, J.C. On maximal monotonicity of bifunctions on Hadamard manifolds, J. Glob. Optim. (2018) [7] Horvath, C.D. Extension and selection theorems in topological spaces with a generalized convexity structure, Ann. Fac. Sci. Toulouse 2 (1993), 253–269. [8] Kim, K.S. Some convergence theorems for contractive type mappings in CAT(0) spaces, Abst. Appl. Anal. vol. 2013, Article ID 381715, 9 pages. doi.org/10.1155/2013/381715. [9] Kim, W.K. Fan-Browder type fixed point theorems and applications in Hadamard manifolds, Nonlinear Funct. Anal. Appl. 23 (2018) 117–127. [10] Kirk, W.A. and Panyanak, B. A concept of convergence in geodesic spaces, Nonlinear Anal. (2007), doi:10.1016/j.na.2007.04.011 [11] Kristály, A. Location of Nash equilibria: A Riemannian geometrical approach, Proc. Amer. Math. Soc. 138(5) (2010) 1803–1810. [12] Kristály, A. Nash-type equilibria on Riemannian manifolds: A variational approach, J. Math.Pures Appl. 101 (2014) 660–688. [13] Kristály, A., Li, C., Lopez-Acedo, G., and Nicolae, A. What do ‘convexitieso imply on Hadamard manifolds? J. Optim. Theory App. 170 (2016) 1068–1074. DOI 10.1007/s10957-015-0780-2 [14] Kumam, P. and Chaipunya, P. Equilibrium problems and proximal algorithms in Hadamard spaces, arXiv:1807.109000vl [math.OC] 28 Jul 2018. [15] Lee, W. Remarks on the KKM theory of Hadamard manifolds and hyperbolic spaces, Nonlinear Funct. Anal. Appl. 20(4) (2015) 579–593. [16] Li, S.-L., Li, C., Liou, Y.-C., and Yao, J.-C. Existence of solutions for variational inequalities on Riemannian manifolds, Nonlinear Anal. 71(11) (2009) 5695–5706. [17] Németh, S.Z. Variational inequalities on Hadamard manifolds, Nonlinear Anal. 52 (2003) 1491–1498. [18] Park, S. Generalizations of the Nash equilibrium theorem in the KKM theory, Takahashi Legacy, Fixed Point Theory Appl. vol.2010, Article ID 234706, 23pp. doi:10.1155 /2010/234706. [19] Park, S. The KKM principle in abstract convex spaces: Equivalent formulations and applications, Nonlinear Anal. 73 (2010) 1028–1042. [20] Park, S. A genesis of general KKM theorems for abstract convex spaces: Revisited, J. Nonlinear Anal. Optim. 4(1) (2013) 127–132. [21] Park, S. Remarks on “Equilibrium problems in Hadamard manifolds" by V. Colao et al., Nonlinear Funct. Anal. Appl. 18(1) (2013) 23–31. [22] Park, S. Review of recent studies on the KKM theory, II, Nonlinear Funct. Anal. Appl. 19(1) (2014) 143–155. [23] Park, S. Generalizations of some KKM type results on hyperbolic spaces, Nonlinear Funct. Anal. Appl. 23(4) (2018) 805–818. [24] Rahimi, R., Farajzadeh, A.P., and Vaezpour, S.M. A new extension of Fan-KKM theory and equilibrium theory on Hadamard manifolds, Azerbaijan J. Math. 7(2) (2017) 88–112. [25] Reich, S. and Shafrir, I. Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990) 537–558. [26] Tang, G.-J. and Huang, N.-J. Existence theorems of the variational-hemivariational inequalities, J. Glob. Optim. (2012) DOI 10.1007/s10898-012-9884. [27] Tang, G.-J., Zhou, L.-W., and Huang, N.-J. The proximal point algorithm for pseudomonotone variational inequalities on Hadamard manifolds, Optim. Lett. 7 (2013) 779–790. DOI 10.1007/s11590-012-0459-7 [28] R. Walter, On the metric projection onto convex sets in Riemannian spaces, Arch. Math., Vol. XXV (1974) 91–98. [29] Z. Yang, Y.J. Pu, Generalized Browder-type fixed point theorem with stronly geodesic convexity on Hadamard manifolds with applications, Indian J. Pure Appl. Math. 43(2) (2012) 129–144. [30] Z. Yang, Y.J. Pu, Existence and stability of solutions for maximal element theorem on Hadamard manifolds with applica- tions, Nonlinear Anal. 75(2) (2012), 516–525. [31] L.-W. Zhou, N.-J. Huang Generalized KKM theorems on Hadamard manifolds with applications, (2009) http://www.paper.edu.cn/index.php/default/releasepaper/ content/200906-669 [32] L.-W. Zhou, N.-J. Huang, Existence of solutions for vector optimization on Hadamard manifolds, J. Optim. Theory Appl. 157(2013), 44–53. DOI 10.1007/s10957-012-0186-3.